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Existence of Solutions in Continuous-time Optimal
Growth Models
Hippolyte d’Albis, Pascal Gourdel, Cuong Le Van
To cite this version:
Hippolyte d’Albis, Pascal Gourdel, Cuong Le Van. Existence of Solutions in Continuous-time Optimal Growth Models. Economic Theory, Springer Verlag, 2008, 37 (2), pp.321-333. �10.1007/s00199-007-0294-8�. �halshs-00177269�
Existence of Solutions in Continuous-time
Optimal Growth Models
∗
Hippolyte d’Albis, Toulouse School of Economics (LERNA),
University of Toulouse
Pascal Gourdel, CES, University Paris 1, CNRS
†Cuong Le Van, CES, University Paris 1, CNRS
September 20, 2007
Abstract This paper studies the existence of solutions in continuous time optimization problems. It provides a theorem whose conditions can be easily checked in most models of the optimal growth theory, including those with increasing returns and multi-sector economies.
Keywords Existence of solutions, optimization, continuous time, optimal growth.
JEL Classification C61, D91, E13.
1
Introduction
The modern theory of economic growth hinges on the analysis of optimal allocations of scarce resources over time. To discuss this issue, infinite di-mensional optimization techniques in continuous time have been extensively
∗The authors thank R. Amir, J.M. Bonnisseau, B. Cornet and S. Demichelis for
stim-ulating comments and suggestions. We are grateful to the referee for her/his valuable comments. The usual disclaimer applies.
†Corresponding author, MSE, 106 Boulevard de l’Hˆopital, 75013 Paris,
used over the last forty years. In most cases the problem may be re-written as the following variational problem:
max Z +∞ 0 G (xt, ˙xt) e−rtdt, s.t. ∀t, ˙xx0 ∈ Rt∈ Γl+ given.t(xt)
This problem has a solution provided thatR0+∞G (xt, ˙xt) e−rtdt exhibits some
upper semi-continuity and the correspondence Γ some compactness. How-ever, in many papers, those properties are not checked and the existence of the solution is simply assumed. The difficulty is specific to continuous time: to be a feasible path, xt should belong to a ball of L1 which is not compact
for the L1 topology. General results have nevertheless been established.
No-tably, Magill (1981) provides an existence theorem for problems with a G concave in its arguments. However, his theorem does not apply to the large body of the literature that consider increasing returns. It is true that if the problem is not concave, the difficulty increases since traditional optimality conditions for the Social Planner problem are necessary but not anymore sufficient. Chichilnisky (1981) and Romer (1986a) hence make an important step by providing an existence theorem for optimal solution that apply to models with increasing returns. While proved in a different way, the condi-tions they proposed require the concavity of G with respect to the one of its arguments which have the highest derivative. Their demonstrations are very nice but the precise set of conditions they propose is rather technical and it is not simple to know if a given economic problem satisfies it. We hence propose an original existence theorem whose conditions are easy to verify in a given economic model. We show this by confronting some well-known models to our existence conditions.
The demonstration we propose extends an initial result of Askenazy and Le Van (1999) about the regularity of the objective function. Then it uses the Dunford-Pettis Criterion to get some sequential compactness for some weak topology and a multi-dimensional version of the Fatou’s Lemma to deduce properties for the limit with respect to this topology.
The paper is organized as follows. In section 2, the problem is presented while our theorem is proved in section 3. In section 4, we show how our theorem can be easily applied to models of neoclassical growth or endogenous growth.
2
The Problem
Consider the following optimization problem (P):
max xt,yt Z +∞ 0 u(xt)e −rtdt, subject to: ∀t ≥ 0, F (yt) ≤ ˙yt≤ G(yt, yt, xt), xt ∈ RC+, yt ∈ RK+, y0 ∈ RK+, r > 0 are given.
In this setting, xtis the control variable, ytis the state variable, ˙ytits
deriva-tive with respect to time (the definition of ˙yt will be more precise in
Re-mark 1). and yt ∈ RK
+ is a given externality. The sets C and K are finite;
for simplicity, (y1, . . . , yK) denote elements of RK and (x1, . . . , xC) elements
of RC.
Assume:
A1. The functions F (respectively G) are continuous on RK
+ (respectively
on RK
+ × RK+ × RC+).
A2. For any j ∈ K, the function Gj is concave with respect to x t.
A3. There exist (bi ≥ 0, Ai > 0, i = 1, . . . , C) and (aj ≥ 0, A′j > 0,
j = 1, . . . , K) such that: ∀t, ∀j ∈ K, yjt ≤ A ′
jeajt and if x, and y satisfy the
following differential constraint
∀t, F (yt) ≤ ˙yt ≤ G(yt, yt, xt), then ∀t,
∀j ∈ K, ytj ≤ A ′ jeajt, and ˙yj t ≤ A′ jeajt, ∀i ∈ C, xi t≤ Aiebit. A4. r > sup {bi, i ∈ C} .
A5. The function u is concave, non decreasing and upper semi-continuous1
from RC
+ into R∪ {−∞} and finite valued on RC++.
Remark 1 Observe that under A3 and A4, the functions yje−ρt belong to
the Sobolev space W1,1(R
+) , while the functions yj are in L1(e−ρtdt), with
some ρ > max{aj : j ∈ K} and xi are in L1(e−rtdt).
1The extended real line R = [−∞,+∞] will always be endowed with its usual topology
Assumption A5 allows for utility functions unbounded from below, such as u(c) = log(c) or u(c) = cσ/σ with σ < 0 that are extensively used in economic
models. Note that this does not imply the continuity of u. Indeed, let us consider in the case where C = 2, and the function
u(c1, c2) = −(c2− 1) 2 2c1 if c1 > 0 and c2 < 1 −∞ if c1 = 0 and c2 < 1 0 if c2 ≥ 1
It is easy to show that u satisfies A5 but is not continuous at point (0, 1). The hessian matrix is equal to
−(c2− 1) 2 (c1)3 (c2− 1) (c1)2 (c2 − 1) (c1)2 −1 c1
on ]0, +∞[×[0, 1[. The determinant is equal to zero and the other eigenvalue is negative.
We can state the following lemma which extends the corresponding result of Askenazy and Le Van (1999).
Lemma 1 Under A5, the function L1 +(e
−rtdt) → R ∪ {−∞} defined by x →
R+∞
0 u(xt)e
−rtdt is upper semi-continuous for the topology σ(L1(e−rtdt), L∞),
and the usual topology on R.
Proof. We will prove that the function x ∈ L1
+(e−rtdt) →
R+∞ 0 u(xt)e
−rtdt
is upper semi-continuous for the L1-topology. Since it is concave, it will
therefore be upper semi-continuous for the topology σ(L1(e−rtdt), L∞
). The proof will be done in several steps.
Step 1
Claim The function takes values in R ∪ {−∞}. P roof Indeed, let x ∈ L1(e−rtdt), and let us define T
0 = {t : u(x(t)) ≥ 0}.
Since, for a > 0, we can choose p(a) ∈ ∂u(a) (the non-empty superdiffer-ential), this allows to write for all t ≥ 0, u(a) − u(x(t)) ≥ p(a)(a − x(t)). Hence,
Z
T0
u(x(t))e−rtdt ≤ (u(a) − ap(a))
Z T0 e−rtdt + p(a) Z T x(t)e−rtdt < +∞
This implies in particular, R0+∞u(x(t))e−rtdt exists in R and is different of
+∞. Step 2
Claim Let x ∈ L1(e−rtdt). Then
Z +∞ 0
u(x(t))e−rtdt ∈ R ⇔ u(x(t)) ∈ L1(e−rt).
P roof Let us define the measurable sets, T0 = {t : u(x(t)) ≥ 0} and T0′ =
{t : u(x(t)) < 0}. It suffice to recall that the finiteness of RT
0u(x(t))e
−rtdt
allows us to write (in R) ( R+∞ 0 |u(x(t))|e −rtdt = R T0u(x(t))e −rtdt −R T′ 0u(x(t))e −rtdt R+∞ 0 u(x(t))e −rtdt = R T0u(x(t))e −rtdt +R T′ 0u(x(t))e −rtdt.
This proves the second claim. Step 3: P roof of the lemma.
Let (xn) be a nonnegative sequence of functions which converge to x in L1.
We want to prove in R that lim supn
R+∞ 0 u(x
n(t))e−rtdt ≤R+∞
0 u(x(t))e −rtdt.
Let us denote M = lim supnR0+∞u(xn(t))e−rtdt. If M = −∞, the proof is
over.
So assume M ∈ R. Let us consider (xnk) be a subsequence of (xn)
such that M = limk
R+∞ 0 u(x
nk(t))e−rtdt. Note that from Step 2, u(xnk) ∈
L1(e−rtdt) for any k large enough. From Theorem IV.9 in Brezis (1987)
(or Theorem 3.12 in Rudin (1987)), one can assume that xn converges to x
pointwise a.e. Let us fix a ∈ R++ and p ∈ ∂u(a). We can introduce the
non-positive functions
∀t, fn(t) = u(xn(t)) − pa + u(a) − pxn(t) ≤ 0.
From Fatou’s Lemma (see Rudin (1987)), we get
lim sup k Z +∞ 0 fnk(t)e −rtdt ≤ Z +∞ 0 lim sup k fnk(t)e −rtdt ≤ Z +∞ 0 lim sup k
(u(xnk(t)) − pa + u(a) − pxnk(t))e
−rt
dt
In view of the pointwise convergence and the upper semi-continuity of u, we deduce lim sup k Z +∞ 0 fnk(t)e −rtdt ≤ Z +∞ 0
This can be equivalently rewritten as lim sup k Z +∞ 0 (u(xnk(t)) − pxnk(t))e−rtdt ≤ Z +∞ 0 (u(x(t)) − px(t))e−rtdt Since limk R+∞ 0 x nk(t)e−rtdt = R+∞ 0 x(t)e
−rtdt, we deduce finally that M ≤
R+∞
0 u(x(t))e
−rtdt and the proof is complete.
Remark 2 If u(0) = 0, then the function x ∈ L1
+(e−rtdt) →
R+∞
0 u(x(t))e −rtdt
is actually continuous for the L1-topology. Indeed, first notice that since u
is non decreasing, u(0) finite implies that u is bounded away from below by
u(0). We can now apply Theorem 10.2 of Rockafellar (1970) in order to
de-duce that u is lower semi-continuous, hence continuous on RC
+. Since xn→ x
in L1(e−rtdt) when n goes to infinity, from Theorem IV.9 in Brezis (1987)
(or Theorem 3.12 in Rudin (1987)), one can assume that xn converges to x
pointwise a.e. Consequently, u(xn) converges to u(x) pointwise a.e. Applying
again Fatou’s Lemma, we get
Z +∞ 0 u(x(t))e−rtdt = Z +∞ 0 lim inf n→+∞ u(x n
(t))e−rtdt ≤ lim inf
n→+∞
Z +∞ 0
u(xn(t))e−rtdt.
We have proved that the function x ∈ L1
+(e−rtdt) →
R+∞
0 u(x(t))e
−rtdt is
lower semi-continuous. From Lemma 1, it is therefore continuous.
3
Existence of Solutions
This section provides our main result:
Theorem 1 Under assumptions A1, . . . , A5, if the value of Problem (P) is finite, then (P) has a solution.
Proof. First note that the variables (xt, yt) will be said feasible if they
satisfy the constraints of Problem (P). Observe that assumptions A3 and
A4 imply that R0+∞u(xt)e−rtdt is uniformly bounded from above, on the set
of feasible controls xt. Let the sequence (xn) satisfy limn→∞
R+∞ 0 u (x
n t) e
−rtdt
= M def= supR0+∞u(xt)e−rtdt over the set of feasible controls. Assumptions
A3 and A4 guaranty that xn are elements of L1
implies that the sequence satisfies the Dunford-Pettis criterion2 and has a
subsequence, denoted also (xn) for simplicity, which converges to x∗
∈ L1 +
for the topology σ(L1(e−rtdt), L∞
). From Lemma 1, R0+∞u (x∗
t) e−rtdt ≥ M.
To end the proof, it remains to show that x∗
is feasible.
Observe first that for a defined ρ > max {aj : j ∈ K} then the associated
sequences (yn, ˙yn) also satisfy the Dunford-Pettis criterion for L1(e−ρtdt). It
can therefore be assumed that they converge respectively to y∗
, z∗
for the topology σ(L1(e−ρtdt), L∞). Then for all t:
Z t 0 ˙yn(s)ds → ϕ(t)def= Z t 0 z∗ (s)ds. In particular, yn(t) → ϕ(t) + y
0, for every almost t. Since ∀t, ∀j, 0 ≤
yn,j(t) ≤ A′
jeajt ∈ L1(e
−rtdt), use Lebesgue Theorem to obtain that yn
con-verges to ϕ + y0 for the strong topology (and consequently for the weak
topology) of L1(e−ρtdt). Thus y∗ = ϕ + y
0 and ˙y∗ = z∗.
Using the multidimensional Fatou’s Lemma (see Appendix), given t, there exist (θtk)k=1,...,K+C+1 such that PK+C+1k=1 θtk = 1, θtk ≥ 0, ∀k, and
( ˙y∗ t, x ∗ t) = K+C+1X k=1 θtk(ς1tk, ς2tk), where (ς1
tk, ς2tk) ∈ ac( ˙ytn, xnt) the set of cluster points of the sequence ( ˙ytn, xnt).
Given (t, k), there will be some increasing function φtk : N → N such that
( ˙yφtk(n) t , x φtk(n) t ) −→ n→+∞(ς1tk, ς2tk).
Then, for all j ∈ K
˙y∗jt = K+C+1X k=1 θtkς1jtk = K+C+1X k=1 θtk lim n→+∞˙y j,φtk(n) t ≤ K+C+1X k=1 θtk lim n→+∞G j(y t, y φtk(n) t , x φtk(n) t ).
Since G is continuous and yn converges pointwise, it yields:
˙y∗j t ≤ K+C+1X k=1 θtkGj(yt, lim n→+∞y φtk(n) t , lim n→+∞x φtk(n) t ) = K+C+1X k=1 θtkGj(yt, y ∗ t, ς 2 tk).
From A2, ˙yt∗j ≤ G j (yt, y ∗ t, K+C+1X k=1 θtkς2tk) = G j (yt, y ∗ t, x ∗ t). Recall that ˙yt∗j = K+C+1X k=1 θtkς1jtk = K+C+1X k=1 θtk lim n→+∞y j,φtk(n) t ≥ K+C+1X k=1 θtk lim n→+∞F j (yφtk(n) t ).
Since F is continuous and yn converges pointwise, it yields:
˙y∗j t ≥ K+C+1X k=1 θtkFj( lim n→+∞y φtk(n) t ) = K+C+1X k=1 θtkFj(yt) = Fj(yt). Thus, (y∗ , x∗
) have been proved to be feasible. Hence M = supR0+∞u(xt)e−rtdt =
R+∞ 0 u (x
∗
t) e−rtdt.
One can use the proof of the previous theorem to get the following corollary. Corollary 1 Assume A1, A2, A3, A5. Then the following finite horizon prob-lem has a solution.
max xt,yt Z T 0 u(xt)e−rtdt subject to: ∀t ∈ [0, T ], F (yt) ≤ ˙yt≤ G(yt, yt, xt), xt∈ RC+, yt∈ RK+, y0 ∈ RK+, r > 0, T ∈ ]0, +∞[ are given.
Remark 3 Let us consider the Social Planner Problem associated with (P).
max xt,yt Z +∞ 0 u(xt)e −rtdt, subject to:
∀t ≥ 0, F (yt) ≤ ˙yt≤ G(yt, yt, xt) = eG(y, y, x),
xt∈ RC+, yt∈ RK+,
y0 ∈ RK+, r > 0 are given,
where eG(y, y, x) = Φ(y)G(y, y, x), Φ is the constant function equal to 1. The Social Planner Problem is of the same type than (P) where G is replaced by
e G.
4
Applications
In all the following examples, the finiteness of the value is obvious.
4.1
The Ramsey model with discounting
Consider the model developed by Cass (1965) and Koopmans (1965):
max ct,kt Z +∞ 0 u(ct)e−rtdt, subject to ∀t ≥ 0, −δkt≤ ˙kt≤ f (kt) − δkt− ct, ct≥ 0, kt≥ 0, δ ≥ 0, k0 > 0, r > 0 are given.
Variables ctand ktdenote respectively the consumption and the capital stock
at time t. It is supposed that the instantaneous utility function u satisfies assumption A5, and that the production function f is concave, continuous, increasing, differentiable, and satisfies f (0) = 0, f′
(∞) = 0.
Define xt= ct, yt = kt, yt= 1, G(yt, yt, xt) = yt(f (yt) − δyt− xt) , F (yt) =
−δyt. Now, check whether Assumptions A1,...,A4 are satisfied.
A1 : It is obvious that F and G are continuous. A2 : It is also obviously satisfied.
A3: Since f′(∞) = 0, for any ε ∈ (0, r) , there exists B such that ∀y ≥ 0,
f (y) ≤ B + εy. It is then easy to show that there exists A′′
> 0, such that ∀t ≥ 0, yt ≤ A′′eεt. Using −δyt ≤ ˙yt, this implies ∀t ≥ 0, xt ≤ Aeεt with
A = B + εA′′, and | ˙y
t| ≤ A′eεt with A′ = max {δk0, B + εA′′}.
A4 : is automatically satisfied.
Observe that the function ke−rt is in W1,1(R
+), and c is in L1(e−rtdt).
4.2
The Ramsey model with endogenous labor
Consider now the following problem:
max
ct,lt,kt
Z +∞ 0
subject to ∀t ≥ 0
−δkt ≤ ˙kt≤ f (kt, 1 − lt) − δkt− ct,
ct≥ 0, kt≥ 0, 0 ≤ lt ≤ 1,
δ ≥ 0, k0 > 0, r > 0 are given.
The variable lt denotes the leisure at time t. It is supposed that the utility
function u satisfies A5 and that the production function f is concave, in-creasing in both arguments, continuously differentiable, and satisfies f (0, l) = f (k, 0) = 0, and f′
(∞, l) = 0.
As in the previous example, one can find some ε in ]0, r[ such that for any t ≥ 0, ct ≤ Aeεt, kt ≤ A′eεt, ˙kt ≤ A′ eεt. Denote x t = (ct, lt), yt = kt,
yt = 1, G(yt, yt, xt) = yt(f (yt) − δyt− xt) , F (yt) = −δyt. It is then easy to
check that Assumptions A1,. . . , A4 are satisfied. In this model, again the function ke−rtis in W1,1(R
+), and c is in L1(e−rtdt).
Obviously, l is in L∞
.
4.3
Endogenous growth
Consider the Romer (1986b) Model. For a given path kt≥ 0, solve:
max ct,kt Z +∞ 0 u(ct)e−rtdt, subject to ∀t ≥ 0 ct+ it≤ F (kt, kt), ˙kt kt = g( it kt), ct≥ 0, it ≥ 0, kt ≥ 0, k0 > 0, r > 0 are given.
Variables it, kt, ctrespectively denote the investment, the stock of knowledge
and the consumption at time t. It is supposed that the function g is concave, increasing, and satisfies g(0) = 0, g(∞) = λ > 0. Assume, moreover, that F is concave, nonnegative, and satisfies F (y, k) ≤ yαkβ for every (y, k) ≥ 0,
and α > 0, β > 0. The function u, as previously, satisfies Assumption A5. Define:
G(y, y, x) = yg(F(y,y)−xy ) when y > 0, F (y, y) − x ≥ 0,
and:
G(y, 0, 0) = 0.
When xt = ct, yt= kt, the problem is then equivalent to:
max ct,kt Z +∞ 0 u(ct)e−rtdt, subject to ∀t ≥ 0, 0 ≤ ˙kt ≤ G(kt, kt, ct), kt≥ 0, k0 > 0, r > 0 are given.
We assume furthermore that (i) supt∈R+hyt
eλt
i
< +∞, (ii) r > λ(α + β).
This condition ensures that our objective function is finite valued. This also implies that the maximal growth rate of the output (equal to λ(α + β)) must be less than the discount rate r.
Now, check whether Assumptions A1,. . . ,A4 are fulfilled.
A1 : It is easy to see that G is continuous. A2 : It is obvious that G is concave in x.
A3 : Since 0 ≤ ˙yt ≤ λyt, it implies that ∀t, yt ≤ k0eλt, | ˙yt| ≤ λk0eλt and
xt ≤ k0k0eλ(α+β)t.
A4 : It is satisfied from the assumption r > λ(α + β) stated above.
In this model, the functions ke−ρt belong to the Sobolev space W1,1(R +),
with ρ > λ, c is in L1(e−rtdt).
4.4
Endogenous growth with human capital
Consider finally the Lucas (1988) Model. For a given human capital path (ht), solve: max ct,θt,kt,ht Z +∞ 0 u(ct)e−rtdt, subject to: ∀t ≥ 0, −δkt≤ ˙kt ≤ kαt (θtht)1−αh γ t − δkt− ct, 0 ≤ ˙ht≤ φ(1 − θt)ht, ct≥ 0, kt≥ 0, 0 ≤ ht, 0 ≤ θt≤ 1, k0 > 0, h0 > 0 are given, r > 0, 0 < α < 1, γ > 0.
Variables ct, θt, kt and ht denote respectively the consumption, the working
time, the physical capital and the human capital at time t. Moreover, it is supposed that function φ is increasing, concave and satisfies φ(0) = 0, φ(1) = µ > 0 while function u is concave and increasing.
Define xt= (ct, θt), yt= (kt, ht), yt = (1, ht), u(x) = v(c), F (y) = (−δk, 0),
G(yt, yt, xt) = ktα(θtht)1−αh γ
t − δkt− ct.
Assume furthermore that (i) supt∈R+hyt
eµt
i
< +∞, (ii) r > µ(1−α+γ)(1−α) .
Obviously, Assumption A5 is satisfied. Now, check A1,. . . ,A4.
A1 : It is obviously true.
A2 : It is immediate since 0 < α < 1
A3 : It follows from the dynamic constraint on human capital that ˙ht
≤ µh0eµt, ht≤ h0eµt. From the constraints on physical capital, one obtains that
kt ≤ h0(1−α) µ(1−α+γ) 1 1−α eµ(1−α+γ)t(1−α) , ct ≤ h0(1−α) µ(1−α+γ) 1 1−α h1−α+µ0 eµ(1−α+γ)t(1−α) , ˙kt ≤ max h 1 1−α 0 (1−α) µ(1−α+γ) α 1−α , δk0 eµ(1−α+γ)t(1−α) .
Condition (ii) ensures that our objective function is finite valued. It also implies that the maximal growth rate of the output (equal to µ(1−α+γ)(1−α) ) must be less than the discount rate r. Thanks to this additional assumption, A4 is satisfied.
In this model, the function he−ρt belongs to the Sobolev space W1,1(R +),
with ρ > µ while c and k are in L1(e−rtdt).
5
Conclusion
This paper has proposed an original theorem to prove the existence of an optimal solution for most macroeconomic problems. The next step would be to prove the existence of an equilibrium with externality y defined en-dogenously. This equilibrium can be viewed as a fixed-point but proving the existence is however a difficult task. Let us introduce the correspondence of “compatibility” which plays a role analogous to the budget correspondence.
∆(y) = {(x, y) | y(0) = y0 (given) and F (yt) ≤ ˙yt ≤ G(yt, yt, xt)}.
We are looking for an externality y which is a fixed-point of the correspon-dence
Λ(y) = arg max y Z +∞ 0 u(xt)e−rtdt | (x, y) ∈ ∆(y) .
Indeed, in Fixed-Point Theory, the main results required the upper semicon-tinuity of the correspondence Λ (here the solution exhibited by our theorem) with respect to the variable (here, the externality y). This part is difficult to establish since there is a lack of lower semi-continuity of the correspondence ∆ which prevents us to apply the Maximum Principle of Berge.
6
Appendix
The original Fatou’s lemma has been generalized to the following proposition (See e.g. Hildenbrand and Mertens, 1971).
Proposition 1 Let (hn) be an integrably bounded sequence of integrable
map-pings from Ω ⊂ R to Rk, which converges weakly to a integrable mapping f :
Ω to Rk. Then, we have
for a.e. a ∈ Ω, f (a) ∈ coLsn{fn(a)},
where Lsn{xn} denotes the set of limit points of converging subsequences of
(xn), and co(Z) the convex hull of the set Z.
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